Dihedral angles

Statement of result

Explanation

0.15 By the angle between two hyperplanes, I mean the following:

• take the unit vectors v and w orthogonal to these hyperplanes,

• take the smallest angle phi such that cos phi equals v dot w.

1.05 In a root system, the vectors orthogonal to the root hyperplanes are the (simple) roots, so we are really interested in alpha dot beta where alpha and beta are roots. Recall that the dot product is the Killing form.

Proof

1.30 Since we're in a root system, if alpha and beta are roots then P alpha of beta equals a half n beta alpha times alpha for some integer n beta alpha, where P alpha of beta is the orthogonal projection of beta onto the line through alpha.

Example:

2.00 In the SU(3) root system shown below, the projection of beta to the line through alpha gives minus a half alpha.

2.50 By Euclidean geometry, we have P alpha of beta equals alpha dot beta over alpha dot alpha, times alpha

3.35 This means that n beta alpha equals 2 alpha dot beta over alpha dot alpha We also have n alpha beta equals 2 alpha dot beta over beta dot beta so n beta alpha times n alpha beta equals 4 time (alpha dot beta) squared over (alpha dot alpha times beta dot beta), which equals 4 cos squared phi.

4.30 where phi is the angle between the roots.

4.50 We have that cos phi lies strictly between minus 1 and 1 because the roots are not collinear, so n alpha beta times n beta alpha equals 4 cos squared phi lives in the set 0, 1, 2, 3 because it's an integer between 0 and 4.

6.05 These possibilities give phi equals 90, 60, 45 or 30 degrees.